If y=a sin x+b cos x (where a and b are const | vedclass

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(D) Given $y=a \sin x+b \cos x$. Differentiating with respect to $x$,we get $\frac{d y}{d x}=a \cos x-b \sin x$. Now,consider the expression $y^2+\lef

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a constant.

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(D) Given $y=a \sin x+b \cos x$. Differentiating with respect to $x$,we get $\frac{d y}{d x}=a \cos x-b \sin x$. Now,consider the expression $y^2+\left(\frac{d y}{d x}\right)^2$: $y^2+\left(\frac{d y}{d x}\right)^2 = (a \sin x+b \cos x)^2+(a \cos x-b \sin x)^2$ $= (a^2 \sin^2 x + b^2 \cos^2 x + 2ab \sin x \cos x) + (a^2 \cos^2 x + b^2 \sin^2 x - 2ab \sin x \cos x)$ $= a^2(\sin^2 x + \cos^2 x) + b^2(\cos^2 x + \sin^2 x)$ $= a^2(1) + b^2(1) = a^2+b^2$. Since $a$ and $b$ are constants,$a^2+b^2$ is also a constant. Therefore,$y^2+\left(\frac{d y}{d x}\right)^2$ is a constant.

If $y=a \sin x+b \cos x$ (where $a$ and $b$ are constants),then $y^2+\left(\frac{d y}{d x}\right)^2$ is

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